A124069 -id:A124069

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A023052

Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

+10
33

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The old name was "Powerful numbers, definition (3)". Cf. A001694, A007532. - N. J. A. Sloane, Jan 16 2022.

Randle has suggested that these numbers be called "powerful", but this usually refers to a distinct property related to prime factorization, cf. A001694, A036966, A005934.

Numbers m such that m = Sum_{i=1..k} d(i)^s for some s, where d(1..k) are the decimal digits of m.

Superset of A005188 (Plusperfect, narcissistic or Armstrong numbers: s=k), A046197 (s=3), A052455 (s=4), A052464 (s=5), A124068 (s=6, 7), A124069 (s=8). - R. J. Mathar, Jun 15 2009, Jun 22 2009

LINKS

Jerome Raulin, Table of n, a(n) for n = 1..345 (terms 1..255 from Joseph Myers)

Encyclopaedia Britannica, Perfect digital invariant, article "Number patterns and curiosities" online since July 26, 1999, revised Aug 25, 2000.

Hans Havermann, Extended table of values for A023052 and A046074

Donald E. Knuth, The Art of Computer Programming, Volume 4, Pre-Fascicle 9B A Potpourri of Puzzles

J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383.

J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]

Eric Weisstein's World of Mathematics, Narcissistic Number

Index entries for sequences related to powerful numbers

EXAMPLE

153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

MATHEMATICA

Select[Range[0, 10^5], Function[m, AnyTrue[Function[k, Total@ Map[Power[#, k] &, IntegerDigits@ m]] /@ Range@ 10, # == m &]]] (* Michael De Vlieger, Feb 08 2016, Version 10 *)

PROG

(PARI) is(n)=if(n<10, return(1)); my(d=digits(n), m=vecmax(d)); if(m<2, return(0)); for(k=3, logint(n, m), if(sum(i=1, #d, d[i]^k)==n, return(1))); 0 \\ Charles R Greathouse IV, Feb 06 2017

(PARI) select( is_A023052(n, b=10)={n<b|| forstep(p=logint(n, max(vecmax(b=digits(n, b)), 2)), 2, -1, my(t=vecsum([d^p|d<-b])); t>n|| return(t==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019

CROSSREFS

Cf. A001694 (powerful numbers: p|n => p^2|n), A005934 (highly powerful numbers).

Cf. A003321, A007532, A014576, A046074, A046761, A053540, A161752.

Cf. A005188 (here the power must be equal to the number of digits).

In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9).

KEYWORD

nonn,base,nice

AUTHOR

David W. Wilson

EXTENSIONS

Computed to 10^50 by G. N. Gusev (GGN(AT)rm.yaroslavl.ru)

Computed to 10^74 by Xiaoqing Tang

A-number typo corrected by R. J. Mathar, Jun 22 2009

Computed to 10^105 by Joseph Myers

Cross-references edited by Joseph Myers, Jun 28 2009

Edited by M. F. Hasler, Nov 21 2019

STATUS

approved

A046197

Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.

+10
26

0, 1, 153, 370, 371, 407

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Suppose n has d digits; then the sum of the cubes of its digits is <= 729d and n >= 10^(d-1). So d <= 5. It is now easy to check that the numbers shown are the only solutions. [Corrected by M. F. Hasler, Apr 12 2015]

This is row n=3 of A252648. - M. F. Hasler, Apr 12 2015

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 153, p. 50, Ellipses, Paris 2008.

G. H. Hardy, A Mathematician's Apology, Cambridge, 1967.

J. Shallit, Number theory and formal languages, in Emerging applications of number theory (Minneapolis, MN, 1996), 547-570, IMA Vol. Math. Appl., 109, Springer, New York, 1999.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 140.

LINKS

Table of n, a(n) for n=1..6.

H. Lehning, La migration des nombres vers le bonheur, Tangente: L'aventure mathématique, pp. 27 No. 108 Jan-Feb 2006 Pole Paris.

FORMULA

A055012(a(n))=a(n); A165331(a(n))=0; subset of A031179. - Reinhard Zumkeller, Sep 17 2009

EXAMPLE

1^3 + 5^3 + 3^3 = 153. 3^3+7^3 +0^3 = 370.

MATHEMATICA

Select[Range[0, 407], Total[IntegerDigits[#]^3]==# &] (* Stefano Spezia, Sep 08 2024 *)

PROG

(PARI) for(n=0, 10^5, A055012(n)==n&&print1(n", ")) \\ M. F. Hasler, Apr 12 2015

CROSSREFS

Cf. A005188, A023052, A046156.

Cf. A165330, A035504, A008585, A165333, A165334, A165335.

Cf. A052455, A052464, A124068, A124069, A226970, A003321.

KEYWORD

nonn,fini,full,base

AUTHOR

Richard C. Schroeppel

STATUS

approved

A252648

Irregular table of perfect digital invariants for n > 1, i.e., numbers equal to the sum of n-th powers of their digits, read by rows.

+10
13

1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 0, 1, 153, 370, 371, 407, 0, 1, 1634, 8208, 9474, 0, 1, 4150, 4151, 54748, 92727, 93084, 194979, 0, 1, 548834, 0, 1, 1741725, 4210818, 9800817, 9926315, 14459929, 0, 1, 24678050, 24678051, 88593477, 0, 1, 146511208, 472335975, 534494836, 912985153, 0, 1, 4679307774

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The third column is listed in A003321. - M. F. Hasler, Feb 16 2015

For a number x >= 10^(d-1) with d digits, the sum of n-th powers of these digits cannot exceed d*9^n. Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10). - M. F. Hasler, Apr 14 2015

LINKS

Table of n, a(n) for n=0..55.

Don Knuth, Table of a(n) for n=0..732

Don Knuth, CWEB program to generate solutions

EXAMPLE

The table starts:

1; (n = 0; 1 = 1^0.)

0, 1, 2, 3, 4, 5, 6, 7, 8, 9; (n = 1)

0, 1; (n = 2)

0, 1, 153, 370, 371, 407; (n = 3, A046197)

0, 1, 1634, 8208, 9474; (n = 4, A052455)

0, 1, 4150, 4151, 54748, 92727, 93084, 194979; (n = 5, A052464)

0, 1, 548834; (n = 6)

0, 1, 1741725, 4210818, 9800817, 9926315, 14459929; (n = 7, A124068)

0, 1, 24678050, 24678051, 88593477; (n = 8, A124069)

0, 1, 146511208, 472335975, 534494836, 912985153; (n = 9, A226970)

The third column corresponds to A003321.

The term 153 is member of the row n=3 because 153 = 1^3 + 5^3 + 3^3. - M. F. Hasler, Apr 14 2015

PROG

(PARI) row(n)={m=1; while(m*9^n>=10^m, m++); for(k=1, 10^m, sum(i=1, #d=digits(k), d[i]^n)==k && print1(k, ", "))}

for(n=0, 7, print1(row(n), ", "))

(Python)

from itertools import combinations_with_replacement

A252648_list = [1]

for m in range(1, 21):

l, L, dm, xlist, q = 1, 1, [d**m for d in range(10)], [0], 9**m

while l*q >= L:

for c in combinations_with_replacement(range(1, 10), l):

n = sum(dm[d] for d in c)

if sorted(int(d) for d in str(n)) == [0]*(len(str(n))-len(c))+list(c):

xlist.append(n)

l += 1

L *= 10

A252648_list.extend(sorted(xlist)) # Chai Wah Wu, Jan 04 2016

CROSSREFS

Cf. A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Cf. A255668 (row lengths).

KEYWORD

nonn,base,tabf

AUTHOR

Derek Orr, Dec 19 2014

EXTENSIONS

I added two links. - Don Knuth, Sep 10 2015

STATUS

approved

A052455

Fixed points for operation of repeatedly replacing a number with the sum of the fourth power of its digits.

+10
11

0, 1, 1634, 8208, 9474

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

This is row n=4 in A252648. - M. F. Hasler, Apr 12 2015

LINKS

Table of n, a(n) for n=1..5.

FORMULA

a(n) = A055013(a(n)). - M. F. Hasler, Apr 12 2015

EXAMPLE

a(2)=1634 since 1^4+6^4+3^4+4^4=1+1296+81+256=1634

PROG

(PARI) for(n=0, 10^5, A055013(n)==n&&print1(n", ")) \\ M. F. Hasler, Apr 12 2015

CROSSREFS

Cf. A023052, A046074, A046197, A052464, A124068, A124069, A226970, A003321.

KEYWORD

base,full,nonn,fini

AUTHOR

Henry Bottomley, Mar 15 2000

STATUS

approved

A052464

Fixed points for operation of repeatedly replacing a number with the sum of the fifth power of its digits.

+10
9

0, 1, 4150, 4151, 54748, 92727, 93084, 194979

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Equivalently, numbers equal to the sum of 5th powers of their decimal digits. Since this sum is <= 9^5*d for a d-digit number n >= 10^(d-1), there cannot be such a number with more than 6 digits. - M. F. Hasler, Apr 12 2015

LINKS

Table of n, a(n) for n=1..8.

G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.

EXAMPLE

a(2) = 4150 since 4^5 + 1^5 + 5^5 + 0^5 = 1024 + 1 + 3125 + 0 = 4150.

PROG

(PARI) for(n=0, 10^6, A055014(n)==n&&print1(n", ")) \\ M. F. Hasler, Apr 12 2015

CROSSREFS

Cf. A023052, A046074, A046197, A052455, A124068, A124069, A226970, A003321.

KEYWORD

base,fini,full,nonn

AUTHOR

Henry Bottomley, Mar 15 2000

STATUS

approved

A124068

Fixed points for operation of repeatedly replacing a number with the sum of the seventh power of its digits.

+10
9

0, 1, 1741725, 4210818, 9800817, 9926315, 14459929

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The sequence "Fixed points for operation of repeatedly replacing a number by the sum of the sixth power of its digits" has just 3 terms: 0, 1, and 548834.

For a d-digit number n >= 10^(d-1), the sum of 7th powers of its digits is <= 9^7*d, therefore these numbers cannot exceed 41205040. - M. F. Hasler, Apr 12 2015

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

1741725 = 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7.

PROG

(PARI) isok(n) = my(d = digits(n)); sum(k=1, #d, d[k]^7) == n; \\ Michel Marcus, Feb 21 2015

(PARI) for(n=0, 41205040, A123253(n)==n&&print1(n", ")) \\ M. F. Hasler, Apr 12 2015

CROSSREFS

Cf. A046197, A052455, A052464, A124069, A226970, A003321.

KEYWORD

base,fini,full,nonn

AUTHOR

Sébastien Dumortier, Nov 05 2006

STATUS

approved

A226970

Fixed points for the operation of repeatedly replacing a number with the sum of the ninth powers of its digits.

+10
7

0, 1, 146511208, 472335975, 534494836, 912985153

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The only six integers equal to the sum of the ninth powers of their digits.

This is row n=9 of A252648. For a d-digit number n >= 10^(d-1), the sum of 9th powers of its digits is <= 9^9*d, therefore n <= 4112105981. - M. F. Hasler, Apr 12 2015

LINKS

Table of n, a(n) for n=1..6.

EXAMPLE

a(3) = A003321(9);

a(4) = 472335975 = 4^9 + 7^9 + 2^9 + 3^9 + 3^9 + 5^9 + 9^9 + 7^9 + 5^9.

PROG

(PARI) is_A226970(n)=n==sum(i=1, #n=digits(n), n[i]^9)

for(n=0, 4112105981, is_A226970(n)&&print1(n", ")) \\ M. F. Hasler, Apr 12 2015

CROSSREFS

Cf. A046197, A052455, A052464, A124068, A124069, A003321.

KEYWORD

nonn,base,fini,full

AUTHOR

Michel Lagneau, Jun 24 2013

STATUS

approved

A255668

Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

+10
3

1, 10, 2, 6, 5, 8, 3, 7, 5, 6, 3, 10, 2, 3, 3, 2, 4, 6, 2, 6, 3, 4, 2, 7, 5, 10, 2, 9, 2, 9, 2, 6, 3, 5, 3, 6, 3, 5, 5, 7, 2, 2, 4, 9, 6, 9, 5, 7, 2, 3, 2, 4, 2, 3, 6, 4, 5, 4, 2, 4, 4, 4, 3, 7, 3, 6, 3, 4, 3, 3, 4, 3, 4, 5, 3, 4, 5, 5, 3, 3, 2, 3, 2, 4, 3, 8, 3, 5, 2, 7, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Row lengths of the table A252648.

For a number with d digits, the sum of n-th powers cannot exceed d*9^n, but the number is not less than 10^(d-1). Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10).

LINKS

Don Knuth, Table of n, a(n) for n = 0..172

Table of a(n) for n=0..172 [From Don Knuth, Sep 09 2015]

FORMULA

a(n) >= 2 for all n > 0, since 0 and 1 are digital invariants for any power n > 0.

EXAMPLE

a(0)=1 because 1 is the only number equal to the sum of 0th powers of its digits.

a(1)=10 because { 0, 1, ... 9 } are the only numbers equal to the sum of their digits (taken to the power 1).

a(2)=2 because 0 and 1 are the only numbers equal to the sum of the squares of their digits.

a(3)=6 because { 0, 1, 153, 370, 371, 407 } is the set of all numbers equal to the sum of the 3rd powers of their digits, cf. A046197.

For more examples, see the table A252648.

MATHEMATICA

Reap@ For[n = 0, n < 6, n++, Sow@ Length@ Select[Range[0, 10^(n + 1)], Plus @@ (IntegerDigits[#]^n) == # &]] // Flatten // Rest (* Michael De Vlieger, Apr 14 2015 *)

CROSSREFS

Cf. A252648, A003321, A046197, A052455, A052464, A124068, A124069, A226970.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Apr 14 2015

EXTENSIONS

a(10)-a(90) from Don Knuth, Sep 09 2015

STATUS

approved

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